Introduction to Asymptotic Theory Definition 4.4 [Almost Sure Convergence]: {Zn}converges to Z almost surely if Pr[z-z=0=1 We denote Zn-0. Almost sure convergence with order n: -The sequence{Zm,n=l,2,···}is said to be of order smaller than na with probability one if n/n0 as noo.This is denoted as n= 0a.s.(na). The sequence [n,n=1,2,.}is said to be at most of order na with prob- ability one if there exists some constant M<oo such that P(Zn/n> M)=0 as n->oo.This is denoted as Zn =Oa.s.(n).In particular,when a=0,Zn=Oa.s.(1)implies that with probability one,Zn is bounded by some large constant for all n sufficiently large. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 21
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 21 Introduction to Asymptotic Theory Definition 4.4
Introduction to Asymptotic Theory Example 4.7 Let w be uniformly distributed on [0,1],and define Z(w)=w for all w∈[0,1]. and Zn(w)=w+wn for w∈[0,1]： Is Zn -Z0? ●Solution:Consider Ac={w∈2：lim|Zn(w)-Z(w)川=0}. n- Because for any given w∈[0，l),we always have lim|Zn(w)-Z(w川=liml(w+w”)-wl m.->o lim w"=0. →00 ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 22
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 22 Introduction to Asymptotic Theory Example 4.7
Introduction to Asymptotic Theory Example 4.7 Let w be uniformly distributed on [0,1],and define Z(w)=w for all w∈[0,1]. and Zn(w)=w+wn for w∈[0,1]： Is Zn -Z0? In contrast,for w=1,we have lim|Zm(1)-Z(1)川=1m=1≠0. m→0 Thus,Ac =0,1)and P(Ac)=1.We also have P(A)=P(w =1)=0. Almost sure convergence is closely related to pointwise convergence (al- most everywhere).It is also called strong convergence. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 23
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 23 Introduction to Asymptotic Theory Example 4.7
Introduction to Asymptotic Theory Lemma 4.4 [Strong Law of Large Numbers (SLLN)for an IID Random Sample: Suppose {Zt}be IID with E(Zt)=u and EZt<o.Then Znaμasn→0o. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 24
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 24 Introduction to Asymptotic Theory Lemma 4.4
Introduction to Asymptotic Theory Lemma 4.5 If Zn -Z 0 then Zn -Z 0. Questions: If s2 Bo2,do we have sB o? ▣ Yes,it follows from the following continuity lemma with the choice of g(s2)= Vs2=5. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 25
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 25 Introduction to Asymptotic Theory Lemma 4.5 Questions: s: